133
38. Three friends, Andy, Berry, and Cassandra, A, B, and C, jointly insured a commercial property in the ratio of 10 : 9 : 6,
respectively. How will an annual premium of $8,000 be distributed among the three of them?
39. Three friends, Alex, Brooks, and Charlie, have decided to invest $4,000, $6,000, and $2,000, respectively, to start a
software development business. If Charlie decided to leave the business, how much would Alex and Brooks have to
pay for Charlie's share if they want to maintain their initial investment ratio?
40. Chuck decided to build a yacht with his two friends Rob and Bob and they invested $9,000, $11,000, and $6,500,
respectively. After the yacht was built, Bob decided to sell his share of the investment to Chuck and Rob. How much
would each of them have to pay if they want to maintain the same ratio of their investments in the yacht?
41. Abey and Baxter invested equal amounts of money in a business. A year later, Abey withdrew $7,500 making the ratio
of their investments 5 : 9. How much money did each of them invest in the beginning?
42. Jessica and Russel invested equal amounts to start a business. Two months later, Jessica invested an additional $3,000
in the business, making the ratio of their investments 11 : 5. How much money did each of them invest in the
beginning?
43. If A : B = 4 : 3 and B : C = 6 : 5, find A : B : C.
44. If X : Y = 5 : 2 and Y : Z = 7 : 6, find X : Y : Z.
4.2 Proportions
Proportions
When two sets of ratios are equal, we say that they are proportionate to each other. In the proportion
equation, the ratio on the left side of the equation is equal to the ratio on the right side of the equation.
Consider an example where A : B is 50 : 100 and C : D is 30 : 60.
Reducing the ratio to its lowest terms, we obtain the ratio of A : B as 1 : 2 and the ratio of C : D as 1 : 2.
Since these ratios are equal, they are equally proportionate to each other and their proportion
equation is:
A:B=C:D
The proportion equation can also be formed by representing the ratios as fractions.
Equating the fraction obtained by dividing the 1st term by the 2nd term on the left side, to the one
obtained by dividing the 1st term by the 2nd term on the right side we get:
A C
B=D
This proportion equation can be simplified by multiplying both sides of the equation by the product
of both denominators, which is B × D.
A C
B=D
Multiplying both sides by (B × D),
A
C
^B # Dh = D ^B # Dh Simplifying,
B
If two sets of fractions are
equal, then the product
obtained by crossmultiplying the fractions
will be equal.
AD = BC
The same result can be obtained by equating the product of the numerator of the 1st ratio and the
denominator of the 2nd ratio with the product of the denominator of the 1st ratio and the numerator of the
2nd ratio. This is referred to as cross-multiplication and is shown below:
A
C Cross-multiplying,
B
D
AD = BC
If 3 terms of the proportion equation are known, the 4th term can be calculated.
Therefore, A : B = C : D is equivalent to A = C .
B D
4.2 Proportions
134
Proportion Equation With Sets of Ratios Having More Than Two Terms
If A : B : C = D : E : F,
If A : B : C = D : E : F
then,
A
=
B
or,
A
=
D
D
B
E
A D
,
= ,
=
E
C
F
C
F
B C
=
E
F
Then this ratio can be expressed as A = D , B = E , and A = D .
B E C F
C F
Cross multiplying leads to AE = BD,
BF = CE, and
AF = CD
A
B
C
The given ratio can also be expressed as, = = . Cross-multiplying leads to the same result.
D E F
The equivalent ratio, A : B : C = D : E : F can be illustrated in a table as shown:
1st Term
2nd Term
3rd Term
A
B
C
D
E
F
and expressed as,A : D = B : E = C : F
A: B:C= D: E:F
A =
B = C
D
E
F
Cross-multiplying, leads to the same result, as shown above
AE = BD, BF = CE, and AF = CD.
i.e., in fractional form,
Example 4.2-a
Solving for the Unknown Quantity in Proportions
Find the missing term in the following proportions:
(i) 4 : 5 = 8 : x
Solution
(i)
(ii) 6 : x = 10 : 25
4 : 5 = 8 : x
Using fractional notation, 4 = 8 or 4 = 5
5
x
8
x
Cross-multiplying,
4x = 40
40
Simplifying,
x= 4
Therefore,
x = 10
(ii)
3
1st Term
2nd Term
4
5
8
x
1st Term
2nd Term
6 : x = 10 : 25
Using fractional notation, Cross-multiplying,
Simplifying,
Therefore,
x
6 10 or 6
=
=
10
25
x 25
150 = 10x
150
x=
10
6
x
10
25
x = 15
(iii) x : 1.9 = 2.6 : 9.88
Using fractional notation,
Cross-multiplying,
Simplifying,
Therefore,
Chapter 4 | Ratios and Proportions
x
1.9
x
2.6
=
1.9 = 9.88 or 9.88 2.6
9.88x = 4.94
4.94
x=
9.88
x = 0.5
1
(iv) 3 : 3 4 = x : 5 4 (iii) x : 1.9 = 2.6 : 9.88
1st Term
2nd Term
x
1.9
2.6
9.88
135
Solution
continued
3
1
(iv) 3 : 3 4 = x : 5 4 15
21
= x:
4
4
Rewriting as an improper
fraction,
3:
Multiplying both sides by 4, 12 : 15 = 4x : 21
Using fractional notation,
12 4x
15 12
=
15
21 or 21 = 4x
Cross-multiplying,
Simplifying,
60x = 252
252
x=
60
Therefore,
Example 4.2-b
1st Term
2nd Term
12
15
4x
21
x = 4.2
Solving Word Problems Using Proportions
Ben can walk a distance of 9 km in 2 hours. Calculate:
1
(i)
The distance (in km) that Ben can walk in 3 hours.
2
(ii) How long (in hours) will it take him to walk 15 km?
Solution
(i)
Calculating the distance in (km):
km : hr = km : hr
9 : 2 = x : 31
2
Rewriting as an improper fraction, Multiplying both sides by 2, Using fractional notation, 9:2=x:
7
2
18 : 4 = 2x : 7
18 = 2x or 18 = 4
2x 7
7
4
1st Term
2nd Term
18
4
2x
7
Cross-multiplying, 8x = 126
Simplifying, x = 126
8
x = 15.75
Therefore, Ben can walk a distance of 15.75 km in 3 1 hours.
2
(ii)
Calculating the time that it will take him to walk 15km:
km : hr = km : hr
9 : 2 = 15 : x
9 15
=
or 2 = 9
2 x
x 15
Cross multiplying, 9x = 30
Simplifying,x = 30
9
x = 3.333333... = 3.33
Therefore, Ben can walk 15 km in 3.33 hours.
Using fractional notation, 1st Term
2nd Term
9
2
15
x
4.2 Proportions
136
Example 4.2-c
Sharing Using Proportions
Andrew (A), Brandon (B), and Chris (C) decide to form a partnership to start a snow removal
business together. A invests $31,500, B invests $42,000, and C invests $73,500. They agree to share the
profits in the same ratio as their investments.
Solution
(i)
What is the ratio of their investments?
(ii)
In the first year of running the business, A's profit was $27,000. What were B's and C's profits?
(iii)
In the second year, their total profit was $70,000. How much would each of them receive from
this total profit?
(i)
Ratio of their investments:
A:B:C
31,500 : 42,000 : 73,500
Dividing each term by the common factor of 100,
315 : 420 : 735
Dividing each term by the common factor of 5,
63 : 84 : 147
Dividing each term by the common factor of 7,
9 : 12 : 21
Dividing each term by the common factor of 3,
3:4:7
Therefore, the ratio of their investments is 3 : 4 : 7.
(ii) A’s profit was $27,000. B’s and C’s profits are calculated using one of the two methods, as follows:
Method 1:
Ratio of Investment =
Substituting terms,
Using fractional
notation,
Ratio of Profit
A:B:C = A:B:C
3 : 4 : 7 = 27,000 : x : y
3 = 27,000
3 = 27,000
and
x
y
4
7
Cross-multiplying,
3x = 108,000
Simplifying,
x = $36,000.00
3y = 189,000
y = $63,000.00
Therefore, B’s profit is $36,000.00.
C’s profit is $63,000.00.
Method 2:
Substituting terms,
Using fractional
notation,
Hence,
Cross-multiplying,
Simplifying,
Ratio of Investment
=
Ratio of Profit
A:B:C
=
A:B:C
3:4:7
=
27,000 : x : y
3
27,000
=
4 = 7`
x
y
2nd Term
3rd Term
3
4
7
27,000
x
y
3
3
= 4 and 27,000 = 7
27,000
y
x
3x = 108,000
3y = 189,000
x = $36,000.00
Therefore, B’s profit is $36,000.00.
Chapter 4 | Ratios and Proportions
1st Term
y = $63,000.00
C’s profit is $63,000.00.
137
Solution
continued
(iii) In the second year, their total profit was $70,000. The profit that each of them would receive is
calculated by using one of the methods, as follows:
Method 1:
Since A, B, and C agreed to share profits in the same ratio as their investments, the ratio of their
individual investments to their individual profit should be equal to the ratio of the total investment
to the total profit.
By adding the ratio of their investments (3 + 4 + 7), we know that the total profit of $70,000 should
be distributed over 14 units. Therefore,
Ratio of Investment
=
Ratio of Profit
A : B : C : Total
=
A : B : C : Total
3 : 4 : 7 : 14
=
A : B : C : 70,000
Substituting terms,
Using fractional
notation,
Cross-multiplying,
3 =
A
14
70, 000
4 =
B
14
70, 000
7 =
C
14
70, 000
14A = 210,000
14B = 280,000
14C = 490,000
A = $15,000.00
Method 2:
B = $20,000.00 C = $35,000.00
Ratio of Investment = Ratio of Profit
A : B : C : Total = A : B : C : Total
3 : 4 : 7 : 14 = A : B : C : 70,000
Substituting terms,
Using fractional
notation,
Hence,
Simplifying,
3 =
A
14
70, 000
4th Term
3
4
7
14
A
B
C
70,000
A = 3 # 70, 000 14
A = $15,000.00
4 = 14
B 70, 000
7 = 14
C
70, 000
14B = 4 × 70,000
14C = 7 × 70,000
B = 4 # 70,000 14
B = $20,000.00
Method 3: Sharing Using Ratios:
A’s share = 3 × 70,000.00 = $15,000.00
B’s share = 4 × 70,000.00 = $20,000.00
2nd Term 3rd Term
3 = 4 = 7 = 14
B
A
C
70, 000
Cross-multiplying,14A = 3 × 70,000
1st Term
C = 7 # 70, 000
14
C = 35,000.00
14
14
C’s share = 7 × 70,000.00 = $35,000.00
14
Therefore, A, B, and C will receive profits of $15,000.00, $20,000.00, and $35,000.00, respectively.
Pro-rations
Pro-ration is defined as sharing or allocating the quantities, usually the amounts, on a proportionate basis.
Consider an example where Sarah paid $690 for a math course but decided to withdraw from the
course after attending half the course. As she attended only half the course, the college decided to
$690
refund half of her tuition fee, (
= $345). As the college calculated the refund amount
2
proportionate to the time she attended the course, we say that the college refunded her tuition fee
on a pro-rata basis.
4.2 Proportions
138
A few examples where pro-rated calculations are used are:
Example 4.2-d
•
When a propery is sold, the property tax paid in advance will be refunded on a pro-rata basis.
•
When an insurance is cancelled before the end of the period for which the premiums were
paid, the amount refunded is calculated on a pro-rata basis.
•
Employees' overtime pay, part-time pay, and vacation time are calculated on a pro-rata basis.
Calculating the Pro-rated Amount of a Payment
Find the pro-rated insurance premium for seven months if the annual premium paid for car insurance is $2,250.
Solution
Ratio of the premiums paid:
Premium ($) : Time (months) = Premium ($) : Time (months)
Substituting terms, 2,250 : 12 = x : 7
2, 250 12
2, 250 x
= , or
=
Using fractional notation,
x
7
7
12
1st Term
2nd Term
2,250
12
x
7
Cross-multiplying,12x = 15,750
15,750
Solving,
x=
12
x = $1,312.50
Therefore, the pro-rated premium for seven months is $1,312.50.
Example 4.2-e
Calculating the Pro-rated Amount of a Refund
Johnson paid $350 for a 2-year weekly subscription of a health journal. After receiving 18 issues of
the journal in his second year, he decided to cancel his subscription. What should be the amount of
his refund? Assume 1 year = 52 weeks.
Solution
Paid for 104 issues (2 × 52) and received 70 issues (52+18); therefore, he should be refunded for 34
issues (104 - 70).
Issues (#) : Cost ($) = Issues (#) : Cost ($)
104 : 350 = 34 : x
104 350
104
= 34 , or
=
Using fractional notation,
34
350
x
x
Substituting terms,
1st Term
2nd Term
104
350
34
x
Cross-multiplying,104x = 34 × 350
34 × 350
Simpifying,x =
104
= 114.423076...
= $114.42
Therefore, his refund should be $114.42.
4.2 Exercises
Answers to odd-numbered problems are available at the end of the textbook.
1. Determine which of the following pairs of ratios are in proportion:
a. 6 : 9 and 14 : 21
b. 5 : 15 and 2 : 8
c. 18 : 24 and 12 : 16
d. 12 : 60 and 6 : 24
2. Determine which of the following pairs of ratios are in proportion:
a. 9 : 12 and 4 : 3
Chapter 4 | Ratios and Proportions
b. 10 : 30 and 8 : 24
c. 14 : 20 and 28 : 42
d. 15 : 12 and 24 : 30
139
For Problems 3 to 6, solve the proportions for the unknown value.
3.
a.
x : 4 = 27 : 36
4.
a.
x : 8 = 6 : 24
5.
a.
x : 18 = 8 : 11
6.
a.
x : 3.65 = 5.5 : 18.25
1
4
b. 24 : x = 6 : 9
c. 5 : 9 = x : 3
b. 3 : x = 18 : 42
3
4
b. 7 : x = 5 : 3
1
5
4
3
d. 1 : 2 = 5 : x
c. 15 : 5 = x : 15
2
5
b. 2.2 : x = 13.2 : 2.5
c. 1 : 4
1
2
=x:2
d. 28 : 35 = 4: x
3
4
c. 4.25 : 1.87 = x : 2.2
d. 1 2 : 2 4 = 1 43 : x
1
1
d. 2.4 : 1.5 = 7.2 : x
7. A truck requires 96 litres of gas to cover 800 km. How many litres of gas will it require to cover 1,500 km?
8. Based on Alvin’s past experience, it would take his team 5 months to complete two projects. How long would his team
take to complete 8 similar projects?
9. Eric paid property tax of $3,600 for his land that measures 330 square metres. Using the same tax rate, what would his
neighbour's property tax be if the size of the house is 210 square metres and is taxed at the same rate?
10. The city of Brampton charges $1,750 in taxes per year for a 2,000 square metre farm. How much would Maple Farms
have to pay in taxes if they had a 12,275 square metre farm in the same area?
11. On a map, 4 cm represents 5.0 km. If the distance between Town A and Town B on the map is 9.3 cm, how many
kilometres apart are these towns?
12. On a house plan, 1.25 cm represents 3 metres. If the actual length of a room is 5.4 metres, how will this length be
represented in the plan in cm?
13. Steve invested his savings in a GIC, mutual funds, and a fixed deposit in the ratio of 5 : 4 : 3, respectively. If he invested
$10,900 in mutual funds, calculate his investments in the GIC and the fixed deposit.
14. The ratio of the distance from Ann’s house to Mark, Jeff, and Justin’s houses is 3 : 5.25 : 2, respectively. If the distance
from Ann’s house to Mark’s is 9.50 km, calculate the distance from Ann’s house to Justin’s and Ann’s house to Jeff ’s.
15. A, B, and C, started a business with investments in the ratio of 5 : 4 : 3, respectively. A invested $25,000, and all three
of them agreed to share profits in the ratio of their investments.
a. Calculate C's investment.
b. If A's profit was $30,000 in the first year, calculate B and C's profits.
c. How much would each of them receive if, in the second year, the total profit was $135,000?
16. A, B, and C formed a partnership and invested in the ratio of 7 : 9 : 5, respectively. They agreed to share the profit in
the ratio of their investments. A invested $350,000.
a. Calculate B and C's investments in the partnership.
b. If the partnership made a profit of $126,000 in the first year, calculate each partner's share of the profit.
c. In the second year, if A made $38,500 in profit from the partnership, how much did B and C make?
17. A, B, and C invested $35,000, $42,000, and $28,000, respectively, to start an e-learning business. They realized that
they required an additional $45,000 for operating the business. How much did each of them have to individually
invest to maintain their original investment ratio?
18. Three wealthy business partners decided to invest $150,000, $375,000, and $225,000, respectively, to purchase an
industrial plot on the outskirts of the city. They required an additional $90,000 to build an industrial shed on the land.
How much did each of them have to individually invest to maintain their original investment ratio?
19. Chris, Diane, and David invested a total of $520,000 in the ratio of 3 : 4 : 6, respectively to start a business. Two
months later, each of them invested an additional $25,000 into the business. Calculate their new investment ratio
after the additional investments.
20. Michael and his two sisters purchased an office for $720,000. Their individual investments in the office were in the
ratio of 5 : 4 : 3, respectively. After the purchase, they decided to renovate the building and purchase furniture, so
each of them invested an additional $60,000. Calculate their new investment ratio after the additional investments.
21. A student pays $620 for a course that has 25 classes. Find the pro-rated refund she would receive if she only attends
5 classes before withdrawing from the course.
22. Megan joined a driving school that charges $375 for 12 classes. After attending 7 classes, she decided that she did
not like the training and wanted to cancel the remaining classes. Calculate the pro-rated refund she should receive.
23. Frank bought a brand new car on August 01, 2014 and obtained pre-paid insurance of $1,058 for the period of August
01, 2014 to July 31, 2015. After 2 months of using the car, he sold it and cancelled his insurance. Calculate the prorated refund he should receive from the insurance company.
4.2 Proportions
140
24. The owner of a new gaming business decided to insure his servers and computers. His insurance company charged
him a premium of $2,000 per quarter, starting January 01. If the insurance started on February 01, how much prorated insurance premium did he have to pay for the rest of the first quarter? (Hint: Quarter of a year is 3 months).
25. If the annual salary of an employee is $45,000, calculate his bi-weekly salary using pro-ration. Assume that there are
52 weeks in a year and 26 bi-weekly payments.
26. Ashley received a job offer at a company that would pay her $2,800, bi-weekly. What would her annual salary be,
assuming that she would receive 26 payments in a year?
27. Charles set up a new charity fund to support children in need. For every $10 collected by the charity, the Government
donated an additional grant of $5 to the charity. At the end of 3 months, if his charity fund had a total of $135,000,
including the Government grant, calculate the amount the charity received from the Government.
28. The tax on education materials sold in Ontario is such that for every $1.00 worth of materials sold, the buyer would
have to pay an additional $0.05 in taxes. If $25,000.00 worth of textbooks were sold at a bookstore before taxes,
calculate the total amount of tax to be paid by the purchasers.
29. A first semester class in a college has 6 more girls than boys and the ratio of the number of girls to boys in the class
is 8 : 5.
a. How many students are there in the class?
b. If 4 girls and 3 boys joined the class, find the new ratio of girls to boys in the class.
30. The advisory board of a public sector company has 10 more men than women and the ratio of the number of men to
women is 8 : 3.
a. How many people are there on the board?
b. If 4 men and 4 women joined the board, calculate the new ratio of men to women.
31. To estimate the number of tigers in a forest, a team of researchers tagged 84 tigers and released them into the forest.
Six months later, 30 tigers were spotted, out of which 7 had tags. How many tigers were estimated to be in the forest?
32. Researchers were conducting a study to estimate the number of frogs in a pond. They put a bright yellow band on the
legs of 60 frogs and released them into the pond. A few days later, 15 frogs were spotted, out of which 5 had bands.
How many frogs were estimated to be in the pond?
4 Review Exercises
Answers to odd-numbered problems are available at the end of the textbook.
1. What is the ratio of a Canadian quarter (25¢) to a
Canadian $5 bill, reduced to its lowest terms?
2. What is the ratio of 12 minutes to 2 hours, reduced
to its lowest terms?
3. Solve the following proportions for the unknown value:
a. x : 9 = 26 : 39
b. 16 : 24 = 12 : x
c. x : 0.45 = 0.16 : 1.20
4. Solve the following proportions for the unknown value:
a. x : 15 = 24 : 36
b. 8 : 14 = x : 35
c. 12.5 : 70 = x : 1.4
5. Which of the following ratios are equal:
a. 6 : 8 and 18 : 24
b. 30 : 25 and 36 : 48
c. 10 : 35 and 14 : 49
d. 24 : 30 and 12 : 18
Chapter 4 | Ratios and Proportions
6. Which of the following ratios are equal:
a. 16 : 20 and 18 : 30
b. 4 : 10 and 10 : 24
c. 35 : 50 and 21 : 36
d. 20 : 16 and 30 : 24
7. If Christina, a graphic designer, receives an annual
salary of $55,000, calculate her weekly salary using
pro-rations. Assume that there are 52 weeks in a year.
8. As the CFO of a technology company, every year,
Tyler would receive 26 bi-weekly payments of $6,000
each. Calculate his monthly salary.
9. The sales tax on an item costing $350.00 is $45.50.
What will be the sales tax on an item costing
$1,250.00?
10. Peter works 5.5 hours per day and his salary per day
is $112.75. At this rate, how much will he receive if
he works 7.5 hours per day?
11. Which is the better buy based on the unit price:
360 grams for $2.99 or 480 grams for $3.75?